Semicrossed products of C*-algebras and their C*-envelopes

Details Semicrossed products of C*-algebras and their C*-envelopes Semicrossed products of C*-algebras and their C*-envelopes

2011-02-10

arxiv.org/abs/1102.2252v2

Mathematics

Operator Algebras

24 pages, corrected typos, revised argument in the proof of theorem 4.3

Scientific paper

Let $\C$ be a $C^*$-algebra and $\alpha:\C\rightarrow \C$ a unital *-endomorphism. There is a natural way to construct operator algebras which are called semicrossed products, using a convolution induced by the action of $\alpha$ on $\C$. We show that the $C^*$-envelope of a semicrossed product is (a full corner of) a crossed product. As a consequence, we get that, when $\alpha$ is *-injective, the semicrossed products are completely isometrically isomorphic and share the same $\ca$-envelope, the crossed product $\C_\infty \rtimes_{\alpha_\infty} \bbZ$. We show that minimality of the dynamical system $(\C,\alpha)$ is equivalent to non-existence of non-trivial Fourier invariant ideals in the $C^*$-envelope. We get sharper results for commutative dynamical systems.

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